Directed immersions for complex structures

Directed Immersions of Closed Manifolds

Given any finite subset X of the sphere S, n ≥ 2, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R whose Gauss map misses X. In particular, this answers a question of M. Gromov.

Fold Maps, Framed Immersions and Smooth Structures

We show that the cobordism group of fold maps of even non-positive codimension q into a manifold N is a sum of q/2 cobordism groups of framed immersions into N and a group related to diffeomorphism groups of manifolds of dimension q + 1. In the case of maps of odd non-positive codimension q, we show that the cobordism groups of fold maps split off (q − 1)/2 cobordism groups of framed immersions.

CR Singular Immersions of Complex Projective Spaces

Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexification. A classification theorem relates equivalence classes of projections to congruence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex pr...

On Local Isometric Immersions into Complex and Quaternionic Projective Spaces

We will prove that if an open subset of CPn is isometrically immersed into CPm, withm < (4/3)n−2/3, then the image is totally geodesic. We will also prove that if an open subset of HPn isometrically immersed into HPm, with m < (4/3)n− 5/6, then the image is totally geodesic.

Minimal Lagrangian isotropic immersions in indefinite complex space forms